Hilbert symbol
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers . It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert (1897, sections 64, 131, 1998, English translation) in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
The Hilbert symbol has been generalized to higher local fields.
Over a local field K whose multiplicative group of non-zero elements is K×, the quadratic Hilbert symbol is the function (–, –) from K× × K× to {−1,1} defined by
The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
The (bi)multiplicativity, i.e.,
for any a, b1 and b2 in K× is, however, more difficult to prove, and requires the development of local class field theory.
The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group , which is by definition
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